POP Steering parameterization

Eddy diffusivity implementation by Bates, Tulloch, Marshall, and Ferrari

Steering Experiments 030 (spinup from Levitus)

Two day spinup from Levitus. Interestingly this case highlights the sensitivity of the various parameters to U velocities which are 0 at day 1 and just partially spun up at day 2. As constructed the Suppression parameter is structured by Zonal velocities and shows the most variablility.

Working from these pieces I tried a little experiment with the 1st day of data where I replaced the U_mean field with the surface ECCO velocity field, holding U_mean constant throughout the 100-2000m layer. The resulting Suppression field looks very much like the Bate's et al. and Kappa shows corresponding improvements in the tropics with the Ecco Velocities providing the zonal structure in the tropics. There is even a hint of the second zonal low just south of the equator.

\(\color{green} {Parameterizing \quad the\quad Eddy\quad Length\quad Scale} \)

NOTES:

  • \(K=u_{rms}∗{\Gamma * \color{red}{L_{eddy}} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\)

  • \(L_{eddy} = min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}})\)

  • Since the two length scales we are using in the parameterization of \(L_{eddy}\) depend on the Baroclinic wave speed \(c_r\) we will first check \(c_r\) against Chelton 1998.

\(\quad\quad\quad\quad\quad\quad\quad\) First Baroclinic Wave Speed \(c_r\) CESM

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\(\quad\quad\quad\quad\quad\quad\) Chelton Baroclinic gravity wave phase speed

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\(\quad\quad\quad\) The barclinic wave speed looks reasonable so lets compare the Paramaterized Rossby Radius to Chelton

\(\quad\quad\quad\quad\quad\quad\) First Baroclinic Rossby Radius CESM \(\quad (L_{eddy})\)

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\(\color{green} {Parameterizing \quad Zonal\quad Eddy\quad Phase\quad Speed}\quad (\color{red}{c})\)

NOTES:

  • \(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - \color{red}{c}|^2 /u_{rms}^2 (z=0)}\)

  • \(\require{cancel}\cancel{c = - \beta * {L_r^2}}\) \(L_r\) too high at equator

  • \(c = - \beta * L_{eddy}^2\)

\(\quad\quad\quad\quad\quad\) Eddy Phase speed CESM

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\(\quad \quad \quad \quad \)Compared to Hughes, the phase speed is much too high near the equator.

\(\quad \quad \quad \quad \)The first tuning mod is to limit the phase speed to 20cm/s.

\(\quad \quad \quad \quad \)Eddy Phase speed CESM \((c = max(- \beta * L_{eddy}^2,-20) )\)

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\(\quad\quad\quad\quad\quad\quad\) Hughes Phase Speed (cm/s) from Tulloch Marshall Smith '09

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\(\color{green}{Zonal\quad Velocity \quad Term}\quad (\color{red}{u_{mean}})\)

NOTES:

  • \(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |\color{red}{u_{mean}} - c|^2 /u_{rms}^2 (z=0))}\)

  • Here \(U_{mean}\) is just CESM surface velocity.

\(\quad\quad \quad \quad\quad \) Zonal Velocity CESM

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48.6693 -67.6657

\(\color{green}{(U-c)\quad Term}\)

NOTES:

  • \(c = max(- \beta * L_r^2,-20)\) )

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) Bates et al

\(\color{green}{(U-c)^2 \quad Term}\)

NOTES:

  • \(c = max(- \beta * L_{eddy}^2,-20)\)

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) Bates et al

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\(\color{green}{Parameterizing \quad Eady \quad Growth \quad Rate}\quad \color{red}{\sigma}\)

NOTES:

  • The original derivation of \(\sigma_{vi}\) goes to 0 at the equator because of \(f\)

\(\quad\quad\quad\quad\quad\sigma_{vi} = {f \over \sqrt{R_i}}\)

  • The new derivation of \(\sigma_{vi}\):

\(\quad\quad\quad\quad\quad R_i = {f^2 N^2 \over {\underbrace{{ {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial y})^2 + {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial z})^2} }_\text{m^4}}}\quad\quad = \quad\quad {f^2N^2 \over m^4} \)

\(\quad\quad\quad\quad\quad \sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)

\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) CESM

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<matplotlib.figure.Figure at 0x7f08a87cba90>

\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) Bates et al

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\(\color{green}{Parameterizing \quad RMS \quad eddy \quad velocity}\quad \color{red}{u_{rms}}\)

NOTES:

  • \(K=\color{red}{u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2(z=0)}\)

  • \(u_{rms} = alpha*{\sigma_{vi}}*L_{eddy}\)

  • alpha (scaling constant) = 4

  • \(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)

  • \(u_{rms}\) is limited to 5 cm/s $ max(u_{rms},5.)$

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity \((u_{rms})\quad\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms} \) Bates et al

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\(\color{green}{u_{rms}^2 \quad Term} \quad\)

NOTES:

  • \(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /\color{red}{u_{rms}^2} (z=0)} \)
  • \(u_{rms}^2\) is a surface value

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity Squared \((u_{rms}^2)\quad\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms}^2 \) Bates et al

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\(\color{green}{{(U-c)}^2 \over {u_{rms}^2}} \quad \color{green}{Term}\)

NOTES:

  • \(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * \color{red}{|u_{mean} - c|^2/{u_{rms}^2} (z=0)}} \)
  • \(c = max(- \beta * L_{eddy}^2,-20)\)
  • \(u_{rms} = max(u_{rms},5.)\)
  • \(u_{rms}^2\) is a surface value

\(\quad\quad\quad\quad\quad\quad\) \({(U-c)}^2 \over {u_{rms}^2} \quad\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {|u-c|^2 \over u_{rms}^2} \) Bates et al

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\(\color{green}{\quad Suppression \quad factor = {1 \over (1 + b1 * |\bar u - c|^2 /u_{rms (z=0)}^2 )}}\)

NOTES:

  • \(c = max(- \beta * L_{eddy}^2,-20)\)
  • \(u_{rms} = max(u_{rms},5.)\)
  • \(u_{rms}^2\) is a surface value
  • \(b1\) (scaling constant) = 4.

\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor CESM

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\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor Bates

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\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) Bates

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\(\color{green} {\quad Parameterizing \quad The \quad Mixing \quad Term \quad }\color{red}{L_{mix}}\)

NOTES:

  • \(\color{red}{L_{mix}} = \Gamma * L_{eddy} * Suppression\)

  • \(Suppression= {1 \over (1 + b1 * |u_{mean} - c|^2 /u_{rms (z=0)}^2 )}\)

  • \(\color{red}{L_{mix}} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\)

  • \(\Gamma = 1.75\) (Tuning mod: the original Gamma of .35 produced a Kappa with the correct structure but too weak.

\(\quad\quad\quad\quad\quad\quad\quad\) LMIX CESM

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\(\color{green}{\quad Parameterizing \quad Eddy \quad Diffusivity \quad} (\color{red}{K})\)

NOTES:

  • \(\color{red}{K}=u_{rms}∗L_{mix}\)

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) Bates

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\(\quad\quad\quad\quad\quad\quad\quad\) (Zonal x Depth) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\) K limited to (100 < K < 10000)

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Here is the Bate's version

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\(\quad\quad\quad\quad\quad\quad\quad\) Zonal average of the N2 normalized scaling CESM

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The alternate derivation of \(\sigma_{vi}\):

\(R_i = {N^2\over{(\frac{\partial u}{\partial z})^2+(\frac{\partial v}{\partial z})^2}}\)

\(N^2={-g \over \rho_0 }\frac{\partial \rho}{\partial z}\)

After hydrostatic and geostrophic approximations

\(f \frac {\partial v}{\partial z} = {{-g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad f \frac {\partial u}{\partial z} = {{g \over \rho_0 }\frac {\partial \rho}{\partial y}} \)

so

\(\frac {\partial v}{\partial z} = {{-1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad \frac {\partial u}{\partial z} = {{1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial y}}\)

\(R_i = {f^2 N^2 \over {\underbrace{{ {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial y})^2 + {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial z})^2} }_\text{m^4}}}\quad\quad = \quad\quad {f^2N^2 \over m^4} \)

\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)

\(RX_1 = RX_{east} = \Delta\rho_x = \rho_{i+1,j} - \rho_{i,j}\)

\(RY_1 = RY_{north} = \Delta\rho_y = \rho_{i,j+1} - \rho_{i,j}\)

\(RZ_1 = RZ_{k+1} = \Delta\rho_z = \rho_{k} - \rho_{k+1}\)

\(\displaystyle{1 \over L_{R_i}} \displaystyle\int_{2000m}^{100m} \left\lbrace { {-g\over\rho_0}{\frac {\partial \rho} {\partial z}} \over { {g^2 \over \rho_0^2 } \left[( \frac{\partial \rho}{\partial y})^2 + ( \frac{\partial \rho}{\partial z})^2\right] } } \right\rbrace dz\)

Note: missing \(f^2\) which will be cancelled when forming \(\sigma_{vi}\)

\(\quad\quad\) so \(\cdots\) this is not \(R_i\)

Implementation notes

Numerator : Top \(= -grav * RZ_{SAVE}(\cdots k+1) * dzwr(k)\)

Denominator :

\(\begin{align} work1 = p25 & * ( RX(..,i_{east},k)^2 \\ & + RX(..,i_{west},k)^2 \\ & + RX(..,i_{east},k+1)^2 \\ & + RX(..,i_{west},k+1)^2 ) / DXT(i,j)^2 \\ \end{align}\)

\(\begin{align} work2 = p25 & * ( RY(..,j_{north},k)^2 \\ & + RY(..,j_{south},k)^2 \\ & + RY(..,j_{north},k+1)^2 \\ & + RY(..,j_{south},k+1)^2 / DYT(i,j)^2 \\ \end{align}\)

\(\begin{align} work3 = {\left( TOP \over (grav^2*(work1+work2))\right)}*dzw(k) \end{align}\)

Notes:

1)Need to be careful at top and bottom of ocean
2)Accurate dzw(k) for each (i,j) to form $L_{R_i}$
3) When constructing $sigma$ itself, use $RZ_{SAVE}$ with a minimum N value
4) use eps2

Posted by John Truesdale